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Holomorphic Almost Modular Forms
Author(s) -
Marklof Jens
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003364
Subject(s) - holomorphic function , mathematics , upper half plane , modulo , modular form , pure mathematics , logarithm , invariant (physics) , congruence (geometry) , modular design , complex plane , discrete mathematics , mathematical analysis , geometry , mathematical physics , computer science , operating system
Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(2,Z). It is proved that such functions have a rotation‐invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of∏ n = 1 ∞( 1 − exp ( 2 π i n 2 z ) ). The paper is motivated by the author's previous studies [ Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence n 2 x modulo one. 2000 Mathematics Subject Classification 11F11 (primary), 11F06, 11J71 (secondary).